of Networked Hand-Held Devices
PROJECT RESULTS TO DATE To date, we will describe our advances in five categories—technological, classroom activity structures, student-change results, teacher-change results, and the refinement of issues needing further investigation.
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Technology advances are based in the following work:
- Modification of our (now commercially available) version of MathWorlds for the TI-83+ graphing calculator to exploit the TI Navigator prototype classroom network in as seamless a way as possible. After log-in to the network, it is possible to send and receive TI-83+ MathWorlds documents (in the form of "application variables") from inside the MathWorlds application using an extension of its menu system.
- Retrofitting a Macintosh-only version of MathWorlds to pre-prototype new aggregation activity structures (described below) to accept and aggregate student functions sent from the graphing calculator to the teacher’s workstation running the Macintosh-only version of MathWorlds.
- Based on classroom tests of the pre-prototype application, modifying the cross-platform Java MathWorlds to support aggregation on a standard computer network, including (a) a MathWorlds Server to collect and distribute Java MathWorlds functions and documents, and (b) means by which a teacher can easily manage the import and display of multiple student functions for pedagogical purposes.
- Modifying the cross-platform Java MathWorlds to support importing of TI-83+ MathWorlds functions and documents into the computer-based MathWorlds in as direct and easy-to-use a way as possible.
- Creating a prototype version of MathWorlds on the Palm to exploit peer-peer beaming and its use in new student interaction activities for the SRI site.
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New Activity Structures: As planned, our focused interventions are generating new activity structures that exploit different varieties of classroom connectivity, and we are testing the viability and impact of those structures. Importantly, each category is highly generative, scalable, and flexible in terms of being applicable to serve a wide variety of curricular objectives. And each applies well beyond the content that we are working with. We will describe these categories, beginning with activities supported by student-student beaming between hand-helds (Palm Pilots were used, although earlier prototyping was done with iMacs).
Peer-Peer Information-Exchange Challenges: The structure of this activity involves one student (A), sending a mathematical object to a second student (B), where B’s view of the object is limited or different from A’s in some important way by the given configuration of the software. B is to guess A’s object in the terms of A’s view based on successive carefully crafted clues from A given in response to incorrect guesses from B. Typically, after a correct guess, A and B exchange roles. This activity structure engenders intense focus on the mathematical issues at hand, two-way translation between mathematical notations and their natural language formulation, and abundant opportunity to fuel purposeful cognitive activity with affectively driven peer-peer engagement.- In one case worked on, A creates a function in an algebraic view and beams it to B who sees it only in a graphical view. Based on his/her interpretation of the graph, B guesses the algebraic form of the function. If not correct, then A offers a carefully designed clue to move B closer to a correct answer, but without giving away the answer. Clearly, Both A and B must work very hard to translate knowledge about the relations between algebraic and graphical views of functions, either into verbal clues (A), or from verbal clues (B). This can be thought of as a graphical version of the classic "Guess My Rule" activity. By varying the functions in type and complexity, this activity can serve a wide range of topics that involve relations between algebraic and graphical views of functions.
- Another frequently used pair of versions of this activity structure involved the relation between a position or velocity vs. time graph of a motion and the motion itself. A makes and sends a position or velocity vs. time graph and sends it to B. B can run the motion determined by the graph, but not see the graph, and must guess the graph. In this case, if the graph is defined piecewise, then B must describe the graph in great detail. The activity engenders intense focus on the relation between a graph and its associated motion, at a highly detailed level.
- A third pair of versions of this activity involved, respectively, A sending a position vs. time graph to B and B seeing only its velocity vs. time graph (so B has to integrate the velocity function), or, A sending a velocity vs. time graph to B and B seeing only its position vs. time graph (so B has to determine the slope–or derivative–of the received position vs. time function).
- A variation on the previous example involves B using the additional information provided by the motion associated with the received position or velocity graph. This helps illustrates how this activity structure is subject to variation and elaboration. One can finely tune, in the software configuration, the information that B has available on which to fashion a guess.
The next series of categories involves individual students or small groups of students sending a mathematical object to the teacher where it is subject to processing of one sort or another, and public display using the teacher’s computational device (most often, but not necessarily a workstation of some kind).
Creating and Sharing a Personally Meaningful Mathematical Object with the Class–Mathematical Performances. This is a relatively simple type of activity, but one that we feel has enormous pedagogical potential because of the ways it taps into adolescent students’ personal identity, their need for recognition, and their creativity in expressing their unique personal experience. It also serves to focus class attention, which leads to opportunity for follow-up engagement by the teacher to exploit issues raised, for pedagogical and curricular purposes. We provide an example with links to the instructional material as well as graphics and student scripts illustrating the activity.- Create an Exciting Sack Race: We provide the graph of a constant velocity position vs. time function which controls the (horizontal) screen motion of one object (A) and ask the student (1) to write a race-script for an "exciting race" with A and (2) to create a position vs. time graph for B that enacts the race. The student then sends the race-data to the teacher who replays the race in front of the class on a large-screen display while the student author of the race "calls the race" by reading their narrative script. The target mathematical content of this activity is the key notion of slope-as-rate-change. This idea is central to using functions to model situations and phenomena and to interpreting graphs of functions of any origin. A secondary content target is the idea of simultaneous conditions underlying simultaneous equations–the basis for developing and solving such equations. We have seen both a large variety of uniquely personal student creations in response to this task and clear indicators of the "performance" aspects of the task–for example, in most cases, the classroom audience breaks into spontaneous applause when the race and story are complete. See here for the teacher’s instructions and a teacher-led example of this activity. This links further to some brief scripts (from student dialog) and animations of students’ scripts and races.
Typically, the class is subdivided into groups, where the size of the group is determined by the teacher or activity designer to fit both the given size of the class and the mathematical activity (so the group might simply be the whole class, or each group might have only two members, meaning students are organized in pairs). Then the students Count-Off inside the group. In this way, each student has a two-number identity that then serves as the value of a "personal parameter" that thus systematically varies across students. The students then create mathematical objects that depend in some critical way on their respective parameter values and then upload these to the teacher where they are aggregated and displayed to the class. We will off a series of examples intending to illustrate the strategy’s flexibility, its power to focus attention, and its power to tap into student personal identity as well as their identity as a member of a group with social dimensions (e.g., as colleagues, classmates, friends, fellow-sufferers, etc.) Again, we offer links to graphic examples drawn from real classroom episodes. To streamline the presentation, let us assume for these examples that students are formed into groups with 3-5 members, so each student has a Count-Off Number ranging from 1 to 5, and the number of groups will depend on the size of the class.
We can vary the Group Number, the Count-Off Number, or both. Assuming members of a group are physically adjacent, then varying the Count-Off Number allows students to see the variation in their group’s productions. On the other hand, if we vary Group Number and not the Count-Off Number, then group members are creating the same object and can help each other. Choice of which to vary depends on the goals of the activity, of course.- Basic Linear Functions—The "Staggered-Start, Staggered Finish Races" (varying a single parameter): In the simplest cases, students make a linear position vs. time Y = mX + b function where either m or b is their Count-Off Number. In the latter, they make a 2 ft/sec motion defined by the position vs. time function Y = 2X + b where "b" is their Count-Off Number. The resulting set of parallel lines and staggered starting points help reveal the invariance of slope (2 in this case), and how the systematically varying y-intercept relates to initial position. A companion activity involves using their Group Number as a starting point, so everyone in a group travels side-by-side, as shown in the linked figure, where we see the screen after 3 seconds of the 5-second race, and the groups are clearly traveling together. Furthermore, the position vs. time graphs of a given group are coincident, while the respective graphs of the 6 groups are all parallel. Lastly, in the third graphic, we can see the equation of each function and hence the parametric variation reflected in the seven values of "b" in Y = 2X + b.
- Linear Functions—The "Staggered-Start, Simultaneous Finish Race": In this activity, one dot (A) starts at 0 m and travels at 2 m/sec for 6 seconds. Each student starts at 3 times their Group Number and is to finish in a tie with A. Here each student in a group is solving the same problem, but may do so in many different ways. Furthermore, since the Group Numbers vary from 1 to 6, the starting points vary from 3 to 18, which means that the slopes of the graphs [see graph] vary from positive, through 0, to negative, with all members of a group traveling together. In particular in the next graphic we can see how the coefficients of X vary, along with b. Note the special case of Group 4 starting at the "finish line" (3 * 4 = 12), having zero velocity, having X coefficient of 0, and having formula given by Y = 0X + 12. This strongly contextualizes Y = 12 in a family of functions in three ways–algebraically, graphically and in terms of motion (where slope as rate of change is likewise in a central role).
- "Constant of Integration": Each student starts at their Group Number & travels at 2 m/s for 5 sec. We display both position and velocity vs. time graphs for the aggregated functions [see graph]. Note that there is only a single, constant velocity vs. time graph visible since they are all coincident, whereas the respective groups have parallel position vs. time graphs.
Our repeated experience with this activity structure convinces us that it has enormous power to energize a class, infusing it with affect, and to focus students’ attention on specific and important mathematical relationships. We begin with a couple of simple examples with linear functions before illustrating some activities involving Position-Velocity connections.-
Linear Functions–Varying Starting Position (Group Number) and Velocity (Count-Off Number): Start at your Group Number & go for 5 seconds at a velocity (whose numeric value is) equal to your Count-Off Number.
(a) Which graph is yours? Explain your reasoning. [see graph]
(b) Based on your motion only, Where Are You? Explain your reasoning.
(c) Which formula is yours? Explain your reasoning.
In versions (a) and (c), respectively, students are must relate the given initial position and velocity information to vertical intercept and slope of the graphs, or the constants in the formulas. In (b) they must relate the given initial position and velocity information to the motion, with the graphs hidden. Note that the teacher has control of what information that is visible to the students, hence can hide the graphs. In this figure, we have displayed all the functions and representational elements simultaneously. However, we could display the motion with "Marks" dropped on a per-second basis, as shown here. The teacher can even scramble the order of the objects as needed. In the previous figure, we have included an "outlier." The potential role of errors is enhanced, although so is the potential for student embarrassment–hence the teacher has the option of not displaying any functions she chooses. We have seen great excitement and excellent logical reasoning occur as students attempt to track down the author of an erroneously produced object. -
Linear Functions–Varying Starting Position (Group Number) and Velocity (Count-Off Number – 2):
Start at your Group Number & go for 5 seconds at a velocity equal to your Count-Off Number – 2.
(a) Which graph is yours? Explain your reasoning. [see graph]
(b) Based on your motion only, Where Are You? Explain your reasoning.
(c) Which formula is yours? Explain your reasoning.
This example illustrates the flexibility in the strategy to manipulate problem features to serve particular objectives. Here, through the device of requiring the velocity to be the student’s Count-Off Number — 2, we introduce zero and negative slopes within the same group, which usually leads to fruitful conversation as students sort out who is who within a group, and provide feedback to one another when an erroneous function is produced. -
Velocity?Position: Make a Velocity Function–Where Are You?:
Starting at zero, make a velocity graph so that you travel at a velocity equal to your Count-Off Number (in m/sec) for 3 seconds, and then for 3 seconds at a velocity equal to your Group Number – 2. We will display your position graph.
(a) Which position vs. time graph is yours? Explain your reasoning. [see graph]
(b) Based on your motion only, Where Are You? Explain your reasoning. [see graph]
This example shares features with the prior examples, but with a focus on velocity-position connections. Worthy of note is the fact that the reasoning that we wish to occur in terms of relating velocity to slope of position graph is exactly what occurs as students try to find themselves in the fan-shaped first segment of the position graphs associated with the velocity graphs that the students construct–and again in the second segment, which provides further branching. Quite clearly, we can vary the conditions as needed to raise a wide range of conceptual issues. For example we can enforce a requirement such as a velocity equal to 2*(-1)Group # where different groups travel in opposite directions. The reader is invited to generate possibilities. -
Dot-Buddies–Adopt-a-Dot: Given a motion for a set of dots whose motion varies systematically in some way, make a motion for yourself so that you move alongside the dot you adopted.
This can be based on making the motion with position or velocity functions, depending on the learning objective. This uses the facility under the control of the teacher to order the sequence of dots on the screen so that an imported dot can be put alongside its adopter. In effect, this turns around the identity, where the original dot on the screen is without identity until a class member "adopts" it. -
Guess My Rule–By Adopting My Graph and My Dot: Half the class makes an algebraic formula (within some family specified in advance, further structured by group or Count-Off Number if desired) and sends it up where its algebraic identity is hidden and it is graphed. The other half of the class is to make formulas which fit the respective graphs that they individually adopt. If needed, the dots can be animated side-by-side.
Clearly, just as with the student-student challenges based on peer-peer beaming outlined above, we can vary the representational givens for pedagogical purposes as needed. And further, hints can be allowed as well–although this is most easily handled using a group structure, where an entire group is responsible for a given target graph and another group is to "guess that group’s rule." This reduces the number of targets to a manageable number–usually 4—6. -
Uploading Student Motions to Create Collective Motions (Marches, dances, story-enactments, …):
(a) Via synthetically defined functions (either graphs or formulas), or
(b) Via physical motions imported using CBR and then uploaded and aggregated, or
(c) Via a combination of synthetic and physically based functions.
Using the MathWorlds capacity to import, aggregate and then animate multiple student-created motions, a wide variety of exciting activities can be produced. These can be done using groups as responsible for a single object per group (e.g., to create a dance of groups), or by varying the objects within a group (e.g., to create a within-group dance). Importantly, the quantitative planning to create motions that relate to other motions is typically exactly the kind of reasoning that we would normally attempt to stimulate in more ordinary activities, except that here the context is strongly social. The common display acts as an arena for the development and testing of ideas in a shared space controlled by the teacher–although subject to control by others as well. As noted, these can be created using any combination of synthetic and physical means. -
Creating Collective Motions Across Different Classrooms or Schools:
While we have not yet done this, we expect that in the coming year we will be able to do versions of the previous activity across different classrooms in the same school or across different schools.
- Student Learning and Achievement:As planned, we utilized focused interventions targeted on particular mathematical content associated with core algebra concepts and skills. One each was done in Spring 2002, based at schools near, respectively SRI International and UMass-Dartmouth. Preliminary pre-post test data from the Massachusetts intervention has been analyzed with very positive results. The course met 5 weeks for 3 afternoons a week in a local high school adjacent to a middle school from which the middle schoolers were drawn. It was taught by a high school teacher, a SimCalc novice teacher "lightly" assisted by two other high school teachers (who mainly handled logistics and field-notes), all trained by SimCalc staff prior to the intervention. Data on the twenty four 7-9th grade students who took both a pre- and post-test show significant gains across each grade level. One third of the sample were volunteers from 7th and 8th grade while the other two thirds were students who had failed or nearly failed the 8th grade required state test and were "strongly recommended " into the course by the department chair. The test consisted of approximately two thirds 10th grade Massachusetts Comprehensive Assessment System items and one third a mix of SimCalc constructed items and items taken from AP Calculus and NAEP items. The statistical results show that all students gained on almost all items, and statistically strong gains summed across items for each of the groups. Of special note were strong gains on open-ended modeling items which most students find especially difficult. We are optimistic that these results can be replicated in subsequent iterations since many of the activities and the technologies were being class-tested for the first time.
